Chapter 13 - The Automation of Proof by Mathematical Induction

Abstract

Inductive inference is a theorem proving using induction rules. It is required for reasoning about objects, events or procedures containing repetition. Also mathematical objects, like the natural numbers, these include: recursive data-structures, like lists or trees; computer programs containing recursion or iteration; and electronic circuits with feedback loops or parameterized components. Many properties of such objects cannot be proved without the use of induction. Inductive inference is thus a vital ingredient of formal methods for synthesizing, verifying, and transforming software and hardware. Induction rules infer universal statements incrementally. The premises of an induction consist of one or more base cases and one or more step cases. There have been two major approaches to the automation of inductive proof: explicit and implicit. The chapter is concerned with explicit induction, in which induction rules are explicitly incorporated into proofs. In implicit induction the conjecture to be proved is added to the axioms. A Knuth–Bendix completion procedure is then applied to the whole system.